Math / Algebra

QuestionSolve the System using the Elimination Method. $\begin{array}{c} -3 x-y=3 \\ x-y=7 \end{array}$
Solution: $\square$ help (points)

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ and $y$ that satisfy both equations simultaneously using the elimination method. Watch out! Be careful with the signs when subtracting equations!
A little slip-up can lead to a wrong answer.

STEP 2

1. Eliminate $y$
2. Solve for $x$
3. Substitute and solve for $y$

STEP 3

Alright, let's kick things off by noticing that both equations have a $-y$ .
This is perfect for the elimination method!
We'll subtract the second equation from the first equation to eliminate the $y$ variable.
Why subtract?
Because $-y - (-y) = -y + y = 0$ .
This is sometimes called "adding to zero".

STEP 4

$\begin{array}{c} -3x - y = 3 \\ -(x - y) = -7 \end{array}$

STEP 5

Distribute the negative sign in the second equation.
Remember, multiplying by $-1$ flips the sign of each term! $\begin{array}{c} -3x - y = 3 \\ -x + y = -7 \end{array}$

STEP 6

Now, let's add the equations together: $(-3x - y) + (-x + y) = 3 + (-7)$

STEP 7

Combine like terms: $-4x = -4$

STEP 8

Divide both sides of the equation by $-4$ to isolate $x$ .
Remember, we're dividing to one, since $\frac{-4}{-4} = 1$ . $\frac{-4x}{-4} = \frac{-4}{-4}$

STEP 9

$x = 1$ We've got our $x$ value!

STEP 10

Now, let's substitute our shiny new $x$ value (which is $1$ ) back into one of the original equations.
Let's use the second equation, $x - y = 7$ , because it looks a little simpler.

STEP 11

Substitute $x = 1$ : $(1) - y = 7$

STEP 12

Subtract $1$ from both sides of the equation: $1 - y - 1 = 7 - 1$

STEP 13

$-y = 6$

STEP 14

Multiply both sides by $-1$ to solve for $y$ : $(-1) \cdot (-y) = (-1) \cdot 6$

STEP 15

$y = -6$ Boom! We found our $y$ value!

STEP 16

The solution is $x = 1$ and $y = -6$ , or $(1, -6)$ .

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QuestionSolve the System using the Elimination Method. $\begin{array}{c} -3 x-y=3 \\ x-y=7 \end{array}$
Solution: $\square$ help (points)

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ and $y$ that satisfy both equations simultaneously using the elimination method. Watch out! Be careful with the signs when subtracting equations!
A little slip-up can lead to a wrong answer.

STEP 2

1. Eliminate $y$
2. Solve for $x$
3. Substitute and solve for $y$

STEP 3

STEP 4

$\begin{array}{c} -3x - y = 3 \\ -(x - y) = -7 \end{array}$

STEP 5

Distribute the negative sign in the second equation.
Remember, multiplying by $-1$ flips the sign of each term! $\begin{array}{c} -3x - y = 3 \\ -x + y = -7 \end{array}$

STEP 6

Now, let's add the equations together: $(-3x - y) + (-x + y) = 3 + (-7)$

STEP 7

Combine like terms: $-4x = -4$

STEP 8

Divide both sides of the equation by $-4$ to isolate $x$ .
Remember, we're dividing to one, since $\frac{-4}{-4} = 1$ . $\frac{-4x}{-4} = \frac{-4}{-4}$

STEP 9

$x = 1$ We've got our $x$ value!

STEP 10

Now, let's substitute our shiny new $x$ value (which is $1$ ) back into one of the original equations.
Let's use the second equation, $x - y = 7$ , because it looks a little simpler.

STEP 11

Substitute $x = 1$ : $(1) - y = 7$

STEP 12

Subtract $1$ from both sides of the equation: $1 - y - 1 = 7 - 1$

STEP 13

$-y = 6$

STEP 14

Multiply both sides by $-1$ to solve for $y$ : $(-1) \cdot (-y) = (-1) \cdot 6$

STEP 15

$y = -6$ Boom! We found our $y$ value!

STEP 16

The solution is $x = 1$ and $y = -6$ , or $(1, -6)$ .

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QuestionSolve the System using the Elimination Method. −3x−y=3x−y=7\begin{array}{c} -3 x-y=3 \\ x-y=7 \end{array}−3x−y=3x−y=7​Solution: □\square□ help (points)

Studdy Solution

STEP 1

What is this asking? We need to find the values of xxx and yyy that satisfy both equations simultaneously using the elimination method. Watch out! Be careful with the signs when subtracting equations!A little slip-up can lead to a wrong answer.

STEP 2

1. Eliminate yyy2. Solve for xxx3. Substitute and solve for yyy

STEP 3

STEP 4

−3x−y=3−(x−y)=−7\begin{array}{c} -3x - y = 3 \\ -(x - y) = -7 \end{array}−3x−y=3−(x−y)=−7​

STEP 5

Distribute the negative sign in the second equation.Remember, multiplying by −1-1−1 flips the sign of each term! −3x−y=3−x+y=−7\begin{array}{c} -3x - y = 3 \\ -x + y = -7 \end{array}−3x−y=3−x+y=−7​

STEP 6

Now, let's add the equations together: (−3x−y)+(−x+y)=3+(−7)(-3x - y) + (-x + y) = 3 + (-7)(−3x−y)+(−x+y)=3+(−7)

STEP 7

Combine like terms: −4x=−4-4x = -4−4x=−4

STEP 8

Divide both sides of the equation by −4-4−4 to isolate xxx.Remember, we're dividing to one, since −4−4=1\frac{-4}{-4} = 1−4−4​=1. −4x−4=−4−4\frac{-4x}{-4} = \frac{-4}{-4}−4−4x​=−4−4​

STEP 9

x=1x = 1x=1 We've got our **xxx value**!

STEP 10

Now, let's **substitute** our shiny new xxx value (which is 111) back into one of the original equations.Let's use the second equation, x−y=7x - y = 7x−y=7, because it looks a little simpler.

STEP 11

Substitute x=1x = 1x=1: (1)−y=7(1) - y = 7(1)−y=7

STEP 12

Subtract 111 from both sides of the equation: 1−y−1=7−11 - y - 1 = 7 - 11−y−1=7−1

STEP 13

−y=6-y = 6−y=6

STEP 14

Multiply both sides by −1-1−1 to solve for yyy: (−1)⋅(−y)=(−1)⋅6(-1) \cdot (-y) = (-1) \cdot 6(−1)⋅(−y)=(−1)⋅6

STEP 15

y=−6y = -6y=−6 Boom! We found our **yyy value**!

STEP 16

The solution is x=1x = 1x=1 and y=−6y = -6y=−6, or (1,−6)(1, -6)(1,−6).

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QuestionSolve the System using the Elimination Method. $\begin{array}{c} -3 x-y=3 \\ x-y=7 \end{array}$
Solution: $\square$ help (points)

What is this asking? We need to find the values of $x$ and $y$ that satisfy both equations simultaneously using the elimination method. Watch out! Be careful with the signs when subtracting equations!
A little slip-up can lead to a wrong answer.

1. Eliminate $y$
2. Solve for $x$
3. Substitute and solve for $y$

$\begin{array}{c} -3x - y = 3 \\ -(x - y) = -7 \end{array}$

Distribute the negative sign in the second equation.
Remember, multiplying by $-1$ flips the sign of each term! $\begin{array}{c} -3x - y = 3 \\ -x + y = -7 \end{array}$

Now, let's add the equations together: $(-3x - y) + (-x + y) = 3 + (-7)$

Combine like terms: $-4x = -4$

Divide both sides of the equation by $-4$ to isolate $x$ .
Remember, we're dividing to one, since $\frac{-4}{-4} = 1$ . $\frac{-4x}{-4} = \frac{-4}{-4}$

$x = 1$ We've got our $x$ value!

Now, let's substitute our shiny new $x$ value (which is $1$ ) back into one of the original equations.
Let's use the second equation, $x - y = 7$ , because it looks a little simpler.

Substitute $x = 1$ : $(1) - y = 7$

Subtract $1$ from both sides of the equation: $1 - y - 1 = 7 - 1$

$-y = 6$

Multiply both sides by $-1$ to solve for $y$ : $(-1) \cdot (-y) = (-1) \cdot 6$

$y = -6$ Boom! We found our $y$ value!

The solution is $x = 1$ and $y = -6$ , or $(1, -6)$ .